Find all solutions to equation: 3tan2x-1=0 I found the solutions to be pi/6 + k(

Question

Find all solutions to equation: 3tan2x-1=0
I found the solutions to be pi/6 + k(pi)   and 7(pi)/6 + k(pi)
is this correct? I am unsure if i left it in one period ornot.
-Also, find all the solutions of the equation in the interval[0,2pi)
cos2x+5cosx +3=0
I cannot figure this one out at all!
3tan2x-1=0
I found the solutions to be pi/6 + k(pi)   and 7(pi)/6 + k(pi)
is this correct? I am unsure if i left it in one period ornot.
-Also, find all the solutions of the equation in the interval[0,2pi)
cos2x+5cosx +3=0
I cannot figure this one out at all!

Explanation / Answer

a few pointers...pi/6 +k(pi) includes 7*pi/6 you have the correct idea that there are several answers first solve for tan (2x) tan(2x) = 1/3 arctan (1/3) = ???? use your calculator that is equal to 2x, so divide by 2, this is x. Now the normal tan curve has a period of pi since you are looking at tan (2x), divide pi by2       (coefficient ofx) so in this instance, the period is pi/2 for one full tan curvecycle. so the solutions repeat every pi/2 your answer for x + k*(pi/2) to look at the solution another way, graph y = 3tan(2x) -1,and look at the x-intercepts (where y = 0) if your window is set for 0, 2*pi, pi/6, 3, -3,1      look at the four solutions... on the second one use the identity, cos 2x = 2(cosx)^2-1 then simplify, factor, set each factor = 0 what values of cos have that value....hint there are 2solutions for one factor, none for the other between 0,2pi

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